Methods and apparatus for linear electric machine

ABSTRACT

An embodiment of a linear electric machine includes two or more phases that define a central bore, and alternating permanent magnets that are disposed within the central bore and are free to move relative the windings. An embodiment of a method for selectively powering the windings is disclosed that enables the machine to realize a commanded force, or to determine the force present by using the current within the windings and the alignment of the magnets relative to the windings.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application Ser.No. 62/461,150 titled “HEMISPHERICAL LINEAR RESPONSE ACTUATOR,” filed onFeb. 20, 2017, and which is incorporated in its entirety herein byreference.

TECHNICAL FIELD

An embodiment relates to the field of Linear Electric Machines (LEMs)and more specifically: those machines that are commutated to produceforces over an arbitrary range of motion; those machines that accuratelypredict and control their force outputs; and those machines that utilizeposition feedback.

SUMMARY

For linear positioning or generating linear forces, a traditional rotaryelectric motor typically is coupled to a screw drive. The motor can beof any technology, be it brushless or brushed, and can include two ormore phases. The coupled screw drive is spun by the motor, and a nut orother threaded mechanism moves up and down the screw drive, producinglinear motion and linear force.

Some approaches include coaxial electrical windings and permanentmagnets. These approaches generate forces through magnetic interactionalone, without relying on a mechanical connection to transform rotarymotion into linear force and motion. Motors including a single coaxialwinding and permanent magnet are often called voice-coil actuators, andoperate in a similar manner to an audio speaker. These devices typicallyaccelerate faster than screw-drive systems, but have characteristicallyshort maximum-travel lengths.

The range limitations of single-winding motors can be addressed byadding multiple phases and commutating their electrical fields similarto brushless rotary motors. When these motors are intended to be stalledor forced to move in some way, position information is required toperform that commutation. Typically, these motors employ alternatingpermanent magnet arrays that are held together mechanically. If thepermanent magnets in the array are spaced closely together—as theytypically are—the resulting force required to keep them from separatingis high, and special construction methods are required. Also, whenalternating magnets are forced together, the resulting total magneticfield in between magnets changes directions and magnitude sharply, asshown in FIG. 4B when taken in contrast to FIG. 3B. This sharp change inmagnetic-field magnitude and direction make it difficult to commutatethe motor such that the force output is constant as the magnet arraymoves relative to the windings; the force ripple typical of thesedevices is sometimes referred to as “cogging” and largely degrades themotors applicability in many applications.

When the force output of any linear motor is important for a motorapplication, a force-sensor of some technology is typically included.The force sensor is often a strain gauge that generates a sense signalthat is amplified to produce a measurable voltage, the magnitude ofwhich depends on the amount of force between the linear motor and itsload. A feedback signal equal to, or derived from, the sense signal isfed back into the system controlling the motor, and the motor iscontrolled to achieve the desired force levels. Generating rapid,controllable forces over a wide range of linear positions, withoutmechanical impedance when unpowered or powered, and with a smooth andlinear force output across the range of travel, has long been acomplicated and unsolved problem.

Existing technologies employing mechanical couplings fail to deliveranything but low-frequency forces, and exhibit a significant amount ofinherent mechanical impedance when unpowered. These technologies alsosuffer from mechanical wear, especially when the output shaft isdynamically acted on by the load. In at least most cases, a loadattached to a shaft of this technology experiences anonlinear-force-response characteristic of the rotary motor generatingthe forces and the screw drive coupling the load to the motor. In theinstances where the force response is actively controlled using a forcesensor and closed-loop control, the system complexity and cost isincreased significantly, both in manufacture and in maintenance.

Existing technologies employing coaxial windings and closely spacedpermanent magnets have rapidly changing magnetic fields and are notcurrently controlled to produce a smooth and linear output force, exceptwhen a force sensor and a closed feedback loop is employed—againincreasing the system cost and complexity.

Therefore, an embodiment solves one or more of the above-discussedproblems by combining physical geometries of motor windings andpermanent magnets with characterization and commutation techniques. Anembodiment allows for construction of a linear electrical machine (LEM)that applies forces through magnetic interaction alone, delivers asmooth and linear force response without requiring a force sensor,effectively converts mechanical energy to electrical energy, andeffectively converts electrical energy to mechanical energy.

The engineer designing a solution while employing an embodimentdisclosed herein will be enabled to control forces rapidly and preciselywhile maintaining a bill-of-materials cost and total system complexitysignificantly lower than he or she would have while employingpreviously-existing technology.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is an isometric view of a four-phase, eight-winding LEM,according to an embodiment.

FIG. 2A is a sectional drawing of the LEM of FIG. 1 at ‘position 0’according to an embodiment. The N-S and S-N labels are intended todemonstrate the polarity of all of the pairs of adjacent permanentmagnets 9A and 9B.

FIG. 2B is a sectional drawing of the same LEM of FIG. 2A moved to‘position π’, according to an embodiment.

FIG. 3A is a cross sectional view of a shaft constructed of permanentmagnets (9A and 9B) and iron spacing material (4)—according to anembodiment. Magnets having parallel hashmarks are understood to havesimilar polarities, denoted by N-S or S-N.

FIG. 3B is an example of the field function according to the shaft shownin FIG. 3A which represents the direction and normalized magnitude ofthe magnetic fields of said shaft along its central axis (5). This graphis scaled such that it aligns horizontally with FIG. 3A.

FIG. 4A is a cross sectional view of a shaft constructed of onlypermanent magnets of alternating polarities—according to an embodiment.

FIG. 4B is an example of the field function according to the shaft shownin FIG. 4A and has a similar relationship to FIG. 4A as FIG. 3B hasrelationship to FIG. 3A.

FIGS. 5A, 5B, 5C, 5D are example Phase Response Functions of the fourphases of the LEM illustrated in FIGS. 1, 2A and 2B—according to anembodiment.

FIGS. 6A, 6B, 6C, 6D are example Phase Commutation Functions of the fourphases of the LEM illustrated in FIGS. 1, 2A and 2B.

FIGS. 7A and 7B show two examples of characteristic commutationfunctions (thin lines), overlaid with characteristic response functions(thick lines)—according to two embodiments.

FIG. 8 s a flow diagram depicting the physical components of anembodiment, the relationships used in the design of those components,and at least one method by which the components are controlled toachieve advantageous effect—according to an embodiment.

FIG. 9A through 9D depict the forces experienced by a magnet duringassembly when magnetic materials are used as spacers, and also depictsthe domains of said magnetic material as said magnet is forced towardthe spacer—according to an embodiment.

FIG. 10A is an isometric of an example two-phase,two-shaft-by-four-winding LEM—according to an embodiment.

FIG. 10B is a sectional view of the LEM of FIG. 10A, illustrating theshaft period (T_(shaft)), winding period (T_(winding)), and phasespatial period (T_(phase))—according to an embodiment.

FIG. 11A is a time-based graph illustrating a shaft moving at constantvelocity after time zero—according to an embodiment.

FIG. 11B is a time-based graph illustrating a commanded force output atvarious points in time—according to an embodiment.

FIG. 11C is a time-based graph illustrating the current within phase Aof a two-phase LEM (solid line), and includes the phase A responsefunction overlaid (dashed line)—according to an embodiment.

FIG. 11D is a time-based graph illustrating the current within phase Bof a two-phase LEM (solid line, left y-axis), and includes the phase Bresponse function overlaid (dashed line, right y-axis)—according to anembodiment.

FIG. 12A is a time-based graph corresponding to FIGS. 11A through 11Dand FIG. 12B illustrating the force output of phase A (solid line, lefty-axis) and phase B (dashed line, right y-axis) of an LEM—according toan embodiment.

FIG. 12B is a time-based graph corresponding to FIG. 11A through 11D andFIG. 12A, illustrating the force output of both phases, and the sum ofthose force output (i.e. the force output of the LEM)—according to anembodiment.

DETAILED DESCRIPTION Conventions

The term ‘current’ is used exclusively to represent electrical current.

The term ‘field’ is used exclusively to represent a magnetic field.

The term ‘shaft position’ is used herein to describe the relativeposition between the shaft and winding pack (both of these elements arediscussed later) in a pack-shaft pair (also discussed later). ‘Shaftspeed’ follows from shaft position in that it describes relative speedbetween the winding pack and the shaft. All descriptions of position andmotion are taken to be relative; it is understood that the shaft, thewinding pack, or both may be changing position and motion with respectto a load or user. A load or user is meant to imply, but not to limit,use of an embodiment for those applications involving applied motion, orthose applications involving applied force: such applications includingrobotic manipulators, production-line manipulators, programmablespring-mass-damper emulators, haptic or emulated-link human-machinecontrols, wave generators (including, but not limited to, waves in soundor water), platform stabilization, and platform motion control.

The term ‘function’ is used exclusively to represent some quantity(e.g., of axial forces, magnetic fields, force-per-amp, oramps-per-force) represented by the y-axis, at a number of positionsalong a spatial dimension represented by the x-axis. While the figuresresemble time-based oscillating functions, it is important to observethat these functions oscillate over a spatial dimension—typically overshaft position. For ease of description, some functions will bedescribed as sinusoid-like. In this context, a function described assuch shares six characteristics with the sine function: it iscontinuous; it is periodic; its integral over a period is zero; itcontains exactly two peak magnitudes of equal and opposite polarity,spaced on the x-axis one-half-period apart and each one-quarter-periodapart from a zero-crossing; it contains exactly two zero-crossings,spaced on the x-axis one-half-period apart and each one-quarter-periodapart from both peak magnitudes' x-axis location; and when translated sothat said function crosses the origin (like a sine function does), saidfunction becomes an odd function, meaning its left-hand-plane is areflection of the right-hand-plane about both the x-axis and y-axis. Allfigures depicting functions herein represent sinusoid-like functions.

The concept of ‘function smoothness’ is discussed throughout thedocument. In this context, two functions' smoothness relative to oneanother can be objectively compared by normalizing the functionsaccording to their maximum magnitudes and finding their derivatives atall positions; the function with lowest maximum derivative magnitude issaid to be smoother. A function's smoothness is said to be improved ifit is changed such that its maximum derivative is reduced—even if indoing so, the average derivative is increased.

The ‘output’ of a device is considered to be a force, or a forcefunction (i.e. a curve representing force outputs at a variety ofpositions) depending on the context. This output is depicted as element21 on flow chart FIG. 8. While significant discussion occurs surroundingthe forces generated, it is recognized that a useful output of anembodiment of the machine described can be the electrical energyconverted from mechanical energy when the shaft position changes.

The term ‘commanded output’ is used to imply some other system iscommunicating with an embodiment, but is not meant to limit or imply thenature of said system. The term “commanded output” is understood torepresent the desired forces generated between the shaft and windings ofan embodiment. The commanded output is often discussed as being constant(i.e. not changing over time), but it is understood that at least mostapplications will involve commanded outputs that change over time.

The term ‘commutation’ is used herein to describe selectively poweringthe phases of an electrical machine to achieve the commanded output.While commutation traditionally aims to achieve a commanded speed,commutation discussed herein aims to achieve a commanded force. It isimportant to note that commutation in the latter sense does notnecessarily result in time-varying currents in the phases; commutationmethods discussed herein involve transforming the position of anembodiment and the commanded output into the currents for the phases.

The term ‘linear force output’ or ‘linear output’ might have threedifferent meanings to someone skilled in the art. Linear output mightdescribe the direction in which force is applied and could be used incomparison to a rotational force output (torque). A linear force outputmight describe the ability of an embodiment to respond linearly to auser command (i.e. realizing twice-as-much output in response totwice-as-much input). A linear output can be used to describe the shapeof the output function; for example, if an embodiment with a perfectlylinear output was subject to a constant user command, the embodiment'soutput would not change if the shaft position were changed. The latterinterpretation should be taken herein; when describing the secondinterpretation, the term ‘linearly proportional to’ is used instead.

The term ‘output ripple’ is used herein to describe the departure of anoutput function from a linear output. For example, consider anembodiment that was said to exhibit output ripple noticeable to a human;a human moving said embodiment with said embodiment being subject to aconstant commanded force, would detect variations in the force output asthe embodiment's shaft position was varied; output ripples in electricmachine is often referred to as “cogging.”

The concept of ‘linearity’ as in ‘output linearity’ is used to comparethe output of an embodiment to a ‘perfectly linear’ output (i.e. afunction having: a perfectly flat curve; the same value for everyposition; a derivative of zero at all points). Electric machinesclaiming a high level of linearity are often referred to as “cog-less.”

Phase Winding Description

Windings of some electrically-conducting medium, surrounded by anelectrically-isolating layer, are used to generate controllable fields.The term ‘windings’ implies one or more turns of the medium. These turnscan be wound beside each other and on top of one another and form aquasi-circular (i.e., spiral) path for electrical charge to flow within,starting at the beginning of the first turn (hereafter referred to asthe ‘positive lead’), and ending at the termination of the last turn(hereafter referred to as the ‘negative lead’). Windings define acentral bore and a central axis about which the turns occur, and anaxial length referred to as the ‘winding length.’ Windings typically,and ideally, have an even spatial distribution of turns throughout theirvolume, although manufacturing processes may cause slight variances inthis distribution. All turns in a winding occur in the same rotationaldirection. When current is passed through the winding, a field isgenerated that is linearly proportional to that current.

Windings are typically copper surrounded by some bonding agent with highdielectric strength. Windings are typically constructed on a windingmachine. The diameter of wire used to construct the windings isdependent on the desired performance characteristics of the machine, anddepends on the operational voltage and other factors. The dielectriccasing of the wire is typically a bondable agent that will soften andadhere to itself upon heating; as part of the manufacturing process,windings are heated so as to form a solid part (i.e., the wires “stuck”together) when cooled.

Windings are combined to form a linear array of windings, referred to asa winding pack. The winding pack includes windings that share a commoncentral axis and a central bore capable of receiving a shaft. FIG. 1illustrates a four-phase, eight-winding winding pack (2), and FIG. 10Aillustrate two, two-phase, four-winding winding packs (17). Acombination of a winding pack and a shaft that is received within thewinding pack is herein referred to as a pack-shaft pair.

All windings within a winding pack are typically constructed with asimilar number of turns and with similar geometry such that all of thewindings are configured to generate a similar magnetic field when a samecurrent is passed through them. Each phase within a winding packtypically contains the same number of windings as all other phases inthe same winding pack. When any two phases within a winding pack containless or more windings than another phase, the commutation for thosephases is scaled appropriately.

It is sometimes advantageous to include a spacing material betweenwindings within a winding pack; when such a construction method is used,winding length equals the winding period less the thickness of thespacing material. The presence of the spacing material can be used toprovide a low thermal-resistance path for heat to travel out of thewindings and into heat spreaders or heat sinks such as a chassis, fins,or a liquid chamber. The spacers can facilitate easier lead routing fromthe windings to the drivers. Spacers can mount sensors, drivers, orother electronics. FIG. 1 illustrates spacers (13), which includewire-grooming and position-sensor mounting features. FIG. 10Aillustrates the use of spacers in a similar manner, but without wiregrooming features. For compact designs, winding spacers can include theelectronic circuits of the windings driver, the phase position sensor,temperature sensors, or other electronics. When an electricallyconductive spacing material is included between windings, it istypically advantageous to include a split, so that electric currentscannot travel in a complete circle around the shaft; this split preventscurrents from being induced in the spacer as the shaft is moved withinit. If currents are allowed to flow through a spacer in response toshaft motion, forces will develop in response to shaft motion as aresult of the magnetic fields induced from said current.

A winding pack's central bore can be configured to receive a centraltube that is fixed to the inner dimension of the windings and/or anyspacing material used; this central tube is used as a sliding interfacefor a shaft, or can be fitted with bushings which act as a slidinginterface. This central tube can be configured to exhibit a high thermalresistance to protect the shaft from the heat of the windings. Someplastics or carbon-fiber materials are suitable material for the windingpack's central tube.

Windings are organized into phases; a phase refers to a winding, or agroup of windings, that can receive electrical power from a singlesource. A phase can consist of several windings connected in series,several windings connected in parallel, or any combination of paralleland series connections. FIGS. 1, 2A, and 2B illustrate a winding packcomprising eight windings organized into four phases (1A through 1D).FIG. 10A illustrates two winding packs each including four windings eachand organized into two phases 60A and 60B.

When phases include windings connected in series, the overall amount ofwiring to the driver will be reduced. However, like the size of wireused in the windings, the connection of windings to produce a phase ineither series, parallel, or a combination of series and parallel dependson the operational voltage and other parameters of the embodiment.

Shaft Description

A shaft is configured to be received within each winding pack's centralbore and configured to have permanent magnetic fields. The term ‘shaft’used herein describe the components that move relative to a windingpack, including but not limited to, permanent magnets, spacing materialbetween those magnets, a container to house said magnets and spacingmaterial, and any other components permanently affixed to this assembly,possibly including a load, a user manipulator, or a mechanical ground.

A shaft's fields are considered permanent in that they are not alteredby normal operation of an embodiment and move instantly with the shaftthrough space. Permanent fields are normally generated by a combinationof permanent magnets and iron, but could be generated by electromagnetsor by other methods.

FIG. 1 shows an example of a shaft received by winding pack 2. Whilethis shaft includes a sleeve (15) for magnets and iron, a sleeve orother container is not necessary. The alternating magnets typicallyincluded in a shaft would normally fly apart due to strong and opposingmagnetic fields so a shaft sleeve can be included to hold them together.The contents of an embodiment's shaft are shown in the sectional viewsof FIGS. 2A and 2B; permanent magnets of alternating polarities 9A and9B are separated by iron slugs 4 and together form a total magneticfield that moves through space with the shaft.

It is convenient to consider a single dimension along the center axis ofthe shaft on which the shaft's fields can be expressed; FIG. 3A includesan illustration of said axis (5) on an embodiment. Typically, all of themagnetic fields along this dimension are directed parallel, orapproximately parallel, to the axis 5. The function created by plottingthe axial field components versus axial position along this dimension isreferred to as the shaft field function; an example of such a functionwith magnitudes normalized to one is shown in FIG. 3B. Typically, theshaft field function is a sinusoid-like function, as described above.The spatial period of the shaft field function is referred to as theshaft period, and relates to an axial distance over which the shaftfield function repeats. The example of FIG. 3B is said to have a shaftperiod of a given the units of measurement of the x-axis (5). While theshaft period is here described in radians, it is understood that thisvalue corresponds to an axial distance; when alternating magnets areused to generate the shaft fields, the shaft period is equal to twicethe magnet-to-magnet spacing.

One way a shaft can be constructed with an appropriate shaft fieldfunction is by locating permanent magnets at a fixed interval and byalternating their polarities. These magnets are polarized such thattheir north and south poles form a line that is parallel to the axis ofthe shaft. FIGS. 2A, 2B, and 10B show cross sections of respectiveshafts constructed of magnets having north-facing-right polarity (9A)and magnets having north-facing-left polarity (9B) separated bylow-carbon iron (4). These magnets can be sized and located such thatthey physically touch one another, or such that a spacing medium (4) canbe used in between them. The resulting shaft period when constructing ashaft in this manner is equal to the twice the fixed interval of thealternating magnets due to the alternating field vectors.

Neodymium permanent magnets are suitable for embodiments herein becauseof their high-density of magnetic fields. When these magnets are chosennot to occupy the full volume of the shaft—typically due to theadvantages that having spaced magnets brings—high-permeability iron,like ‘soft iron’ or low-carbon iron provides good cost-to-performanceresults. The optimal ratio of neodymium to iron in an embodiment willdepend on the windings' construction, the strength of the permanentmagnets, and the desired performance of the embodiment.

A thin shaft sleeve made of a material with high thermal resistance andlow friction is a suitable method of encapsulating the magnets and iron;examples of this material are a carbon fiber or plastic depending on thebushings in which they will travel. This sleeve is often a metal likealuminum or stainless steel due to the extra rigidity it provides. Thisshaft sleeve is not necessary when an embodiment travels on some otherlinear guide mechanism, but can prove useful in deflecting heat from thewindings away from the magnets and further prove useful in ensuring themagnets stay aligned and in place. Thermal protection of the shaft isimportant in embodiments using permanent magnets to generate the shaftfields, because these materials can only operate below certaintemperatures (commonly referred to as the Currie temperature) withoutpermanently (and negatively) altering the magnetic field they generate.Since an unavoidable byproduct of current though a device's windings isheat generated (through resistive power losses), winding temperaturemust be allowed to rise during operation. A good thermal barrier, ormultiple thermal barriers between the windings and shaft allow higherwinding temperatures during operating. This is advantageous as theamount of power dissipated by a heat spreader or absorbed by a heat sinkis proportional to the temperature of said spreader or sink; a devicethat can dissipate more heat (i.e. sustain hotter windings) can supporthigher duty-cycles or sustained operations.

Manufacturing or material non-ideologies may produce variances betweenshaft periods without a noticeable effect on performance. In thiscontext, variance between any two shaft periods can be calculated bycomparing the shaft field values of the two periods at every measurablephase angle within those periods; the maximum variance between these twoperiods is said to be the greatest difference of any two values comparedthis way; the maximum variance of a shaft is said to be the greatestmaximum variance between any two periods within the shaft. The tolerancerequired between periods of the shaft field function will be a functionof the desired output linearity. Two examples follow. The embodimentdepicted in FIGS. 1, 2A and 2B is intended to interact with a human.This embodiment can output 202 newtons, and has a maximum shaft varianceof six-percent of the commanded output, for any commanded output; thisleads to a variation in output of six-percent across the embodiment'stravel, but ultimately this is not detectable to the human operator. Ifanother embodiment intended for remotely controlling surgery implementsspecifies outputs within one-half-a-percent of the command output, itwill warrant (among other things) that greater controls in materialselection and manufacturing to ensure the shaft field function varies byless than one-half-a-percent between any two periods.

Positions and Alignment Description

It is convenient to consider the relative position of the shaft fieldfunction with respect to the center of a winding; this is referred to asthe ‘phase position’ of that winding. Because of the periodic nature ofthe shaft field function, the phase position is also periodic. The phaseposition can be used to represent positions along the central axiswithin a single shaft period. When phase positions are described hereinthey are expressed in radians as values ranging from 0 to 2n. The phaseposition of a winding is said to be zero if the at the center of thewinding, the shaft field function is at its positive peak value. Phase A(1A) of FIG. 2B is said to be at phase position zero. The phase positionchanges when the shaft and winding pack move relative to one another.FIG. 2A illustrates Phase A (1A) at phase position π, Phase B (1B) atphase position 7π/4, Phase C (1C) at phase position π/2, and Phase D(1D) at phase position 5π/4. FIG. 2B illustrates Phase A (1A) at phaseposition zero, Phase B (1B) at phase position 3π/4, Phase C (1C) atphase position 3π/2, and Phase D (1D) at phase position π/4. Note thateach phase in this example four-phase embodiment is separated by a phaseposition of 3π/8 which can be calculated by Equation 1; the importanceof this phase-angle relationship is further explained later.

The phase position of any two windings will differ from each otheraccording to their distance from each other along the linear array ofwindings included in a winding pack. The phase position differencebetween adjacent windings, or between the first and last winding in awinding pack, can be calculated using Equation 3.

FIG. 2A is an example one extreme shaft position; if the shaft were tomove further to the right, the linearity of the device would begin torapidly degrade. The same is true for shaft movement to the left whenthe rightmost magnet is center aligned to the rightmost winding.

In some embodiments, even the extreme positions described above mayresult in excess output ripple near these extreme positions. This isusually negligible, as typical commutation dictates that the outer mostcoil is not receiving power in this condition. However, when anembodiment's output suffers from an unacceptable reduction in linearityat the discussed extreme positions (usually when the magnet length isrelatively short when compared to the shaft period), more magneticmaterial can be added to the shaft; in other words, the extreme positioncan be moved out by some amount (e.g. by a quarter-shaft-period) torestore the required linearity.

The following method is used herein to represent shaft position: a shaftposition of zero indicates that the shaft is at one extreme position,and shaft position is represented by radians where a distance of acorresponds to a distance of one shaft period. An example of shaftposition zero is shown in FIG. 2A. Shaft position will equal a after theshaft (or winding pack) has moved one shaft period away from positionzero. FIG. 2B is said to be at position π, as it has moved one-halfperiod from position zero.

The maximum shaft position is a function of the number of shaft periodswithin the shaft, the number of windings within the winding packreceiving the shaft, and the definition used for the extreme shaftpositions. If the first proposed method of defining extreme positions isused, the embodiment represented by FIGS. 2A and 2B is said to have amaximum shaft position of 11π/2 while the embodiment represented by FIG.10B can be said to have a maximum shaft position of 3π/2.

Parallel Geometries Description

Embodiments include at least one pack-shaft pair. Embodiments includingtwo or more pack-shaft pairs typically: share the same number of phases;individually satisfy the relationships of Equation 2; and include shaftsthat have the same number of periods in their shaft field function asshafts included in all other pairs.

For a given embodiment having more than one pack-shaft pair, amechanical link is typically between all shafts and another mechanicallink is between all winding packs. Pairs included in an embodiment canhave different dimensions from one another; for example, one pack-shaftpair could have a winding period that was half the winding period ofanother pack-shaft pair, so long as the shaft period of the formerpack-shaft pair was half that of the latter. This scale factor betweenpack-shaft pairs is important when considering how the shafts andwinding packs of an embodiment are mechanically linked. These mechanicallinks are such that any relative motion experienced by one shaft-windingpack pair is also experienced, at a scaled amount, by all othershaft-winding pack pairs. This scaled value between any two pack-shaftpairs is, ideally, identical to the ratio of shaft periods, orequivalently the ratio of the winding periods, between the shafts orbetween the winding packs included in the pair respectively. FIGS. 10Aand 10B show one embodiment having multiple two-phase pack-shaft pairs(17); note that the mechanical link between the shafts is notillustrated, and since the dimensions for both packs are equal, themechanical link (15) between the winding packs is one-to-one (rigid)ratio. The mechanical links are typically formed such that when onepack-shaft pair is at an extreme position, so too are all othershaft-pack pairs.

When shafts are mechanically linked in the manner described above, andwinding packs are also mechanically linked in the manner describedabove, the phase position of all phases will be equal for all pack-shaftpairs, regardless of any (scaled) differences in their construction; itfollows then that phase position (which is related to an axial position)measures different distances for any two pack-shaft pairs of differingshaft periods.

Cross-Sectional Relationships

Typically, the windings form a hollow circular cross section on a planeperpendicular to the central axis, and the shaft (including all magneticmaterial used) forms a solid circular cross section on that same plane.Windings configured to form a hollow circle cross section may receive ashaft forming a smaller hollow circular cross section; such a hollowshaft could receive wires, pipes, sensors or other things. Othercross-sectional shapes can be used. For example, windings may form ahollow square cross section, and receive a shaft that forms a solidsquare cross section. Such a configuration would prevent the shaft fromrotating freely within the winding pack.

Axial Relationships

The axial spacing of windings within a pack is referred to as thewinding spatial period, or simply as ‘winding period.’ FIG. 10Billustrates the distinctions between winding period (Twinding), phasespatial period (Tphase), and shaft period (Tshaft) for a two-phaseembodiment. Windings within a winding pack may be touching one another,or may be spaced apart (the winding length can be equal to or less thanthe winding period); in any case windings are typically rigidly securedtogether.

When any phase in a winding pack includes multiple windings, thesewindings are spaced at fixed axial intervals known as the phase spatialperiod or simply as the phase period; the phase period is equal to thewinding period times the number of phases.

When phase position is represented in radians, multiples of a can besubtracted from a winding's phase position. Equation 3 can be used toshow that: if an embodiment includes an even number of phases, windingsthat are separated by an odd integer multiple of the phase period willhave phase positions differing by 7C, while windings separated by eveninteger multiples of the phase spatial period will not differ in phaseposition; and if an embodiment includes an odd number of phases, everywinding within a phase shares a phase position.

Typically, when windings sharing a phase have phase positions differingby 7C (or one-half shaft period), these windings are wired with oppositepolarity; in other words, when current is passed through this phase, itwill travel in opposite directions for windings differing in phasepositions by 7C; windings within a phase are configured this way so thatthe force they generate due to interactions with the shaft sum together,as they would cancel each other otherwise.

If construction of an embodiment benefits from separation of windingswithin a winding pack, windings can be moved to other locations in thewinding pack, so long as they are located axially an integer multiple ofthe phase spatial period from all other windings sharing the same phase.

Driver Description

Phases are selectively and variably provided power through an electriccircuit referred to as a ‘driver.’ There is at least one driver perphase. Drivers are configured to provide power to a phase's windings inboth current-flow directions, and are configured to vary that power witha reasonable resolution—for example, with at least 256 levels (8 bits)per direction.

An H-Bridge circuit that is configured to switch a direct-current (DC)supply onto the coils is an embodiment of the driver circuit. Theswitching frequency of the driver circuit can be over 20 kHz to preventthe generation of an audible noise or “hum.” This driver circuit istypically controlled by a microcontroller running software or firmwareto realize programmed commutation patterns in response to commandedforce outputs.

Phase Position Sensing

Embodiments achieving linear output through the commutation methodsdiscussed below make use of the phase position for every phase included.Thus, an embodiment includes a sensor capable of detecting the phaseposition of all the phases included. Often, this is done by obtaining aphase position for a single winding (and accordingly for all otherwindings sharing a phase) and determining the remaining phases' phaseposition by using the equation of Equation 3; when using the embodimentrepresented by FIG. 2A as an example, if the phase position for phase A(1A) was measured (in this case to be 7 c), phase position for phase B(1B) would be calculated as π+3π/4=7π/4, phase C (1C) would becalculated as π+2*3π/4=5π/2 (which is equivalent to π/2), and phase D(1D) would be calculated as π+3*3π/4=13π/4 (which is equivalent to5π/4).

Examples of suitable phase position sensor are linear position sensorssuch as an optical encoder that scan a code strip, linearpotentiometers, echo or laser sensors; examples are also rotary positionsensors provided they are appropriately coupled to the linear motion ofan embodiment; an example is also an array of hall sensors that canmeasure the shaft's magnetic field while rejecting noise from thewindings; an illustration of such a sensor (14) appears in FIGS. 1 and10A. Shaft-Winding Interaction

Force Response Description

Passing current through a winding in an assembled embodiment will resultin magnetic interaction that produces a force between the winding and ashaft. The magnetic field, and therefore the force that is generated, isdirectly proportional to the current passed through the winding; if thedirection of current is reversed, so too is the force. This linearlyproportional relationship is referred to as a ‘winding force constant’or simply as a ‘force constant.’ The force constant is considered atransfer function which transforms current into force.

The force constant is similar to another linearly proportionalrelationship referred to as a ‘winding generation constant’ or simply‘generation constant.’ The generation constant is considered a transferfunction which transforms shaft speed into a voltage (and in turn, acurrent) induced in the winding.

It is convenient to consider the functions generated by plotting thewinding force constant, and similarly by plotting the winding generationconstant, across shaft positions for an arbitrary winding. Thesefunctions are referred to as a ‘winding response function’ and a‘winding generation function’ respectively. FIGS. 5A, through 5D couldbe examples of winding response functions.

If the winding response function for every winding in an embodiment isknown, if the current in every winding is known, and if the phaseposition is known, the resulting force of the output can be obtained inthe following manner: for each winding in the embodiment, multiply thecurrent in the winding by the winding response function (using phaseposition as argument), which yields the force generated by that winding;sum all the forces to obtain the net force of the embodiment.

Due to the sinusoid-like nature of the shaft field function, the windingresponse functions are also sinusoid-like. This can be seen by analyzingthe winding force constants for various phase positions. First it isimportant to note that when a magnet is axially centered within awinding, the force constant is zero; that is, no amount of current inthe winding will generate force. For example, phase A (1A) of FIG. 2B,which is at phase position zero, cannot produce forces at that shaftposition. This is because the shaft field function is equal in magnitudeand direction for equal distances to the left and right of the axialcenter of windings in phase A; the result of generating a field fromphase A is to pull the shaft on either side of the winding toward thewinding with equal force, or to push the shaft on either side of thewinding with equal force—in either case producing no net force. As theshaft moves relative to the phases, the balance of magnetic field oneither side of phase A (1A) shifts such that one side will have more netpositive field than the other. This imbalance results in a net forceproduced when current is passed through the winding. The imbalance ofmagnetic fields on either side of phase A increases up until phase A'sphase position is π/4 or −π/4 depending on the direction moved; at thisposition, the imbalance of fields is at a maximum (the shaft fieldfunction crosses zero and changes signs at the same location as theaxial center of the windings in phase A). Where a perfect balance offield on either side of a winding's axial center produces a forceconstant of zero, a maximum imbalance of field produces a peak forceconstant (the sign of the force constant will depend on how theimbalance relates to the polarity of the phase). In any case, because ofthe sinusoid-like nature of the shaft fields, the force constant will:increase from zero at a phase position of zero to a peak magnitude at aphase position of π/2; decrease from this peak to zero as the phaseposition is advanced to it; increase to a peak magnitude of oppositepolarity and equal magnitude as at π/2 as the phase position is advancedto 3π/4; and finally decrease in magnitude again to zero as the phaseposition is advanced to 2π. The winding response function is thussinusoid-like.

In some cases, for example if the winding spacing were to be much longerthan the winding length, or if the shaft field function was shaped suchthat it was not sinusoid-like as defined herein, it is possible that thewinding response function is not sinusoid-like; these geometries are nottypical embodiments, and embodiments making reasonable use of thematerials used in their construction will produce sinusoid-like windingresponse functions.

One important relationship between winding response functions andwinding generation functions is that they are linearly proportional toone another. For a winding, energy transformations in both directions(i.e. from electrical energy to mechanical energy and from mechanicalenergy to electrical energy) rely on the density of magnetic fields inthe vicinity of the winding. In fact, both phenomena are linearlyproportional to the density of magnetic fields; that is to say that ifthe magnetic fields in the vicinity of a winding are scaled by somenumber—for example, if they are doubled—then forces generated by currentthrough that winding are in turn doubled, and voltage generated bymoving those magnetic fields is likewise doubled. Because the shaft isthe source of magnetic fields in the vicinity of the windings, the shaftis what dictates how the windings produce force in response to currentand how the windings produce voltage in response to shaft movement. Forthis reason, when the shaft is in a position such that the forceconstant for a winding is zero (i.e. phase position zero), then thewinding's generation constant is zero; likewise, when the shaft is in aposition such that the force constant is maximized, so too is thegeneration constant. Finally, when the shaft is moved to a position suchthat the force constant is doubled, so too is the generation constantdoubled. Another convenient way of illustrating this relationship is bynormalizing the winding response function and the winding generationfunction according to their maximum magnitudes, and plotting them on thesame chart; when this is done, the two functions perfectly overlap. Therelationship between response functions and generation functions isimportant when describing the damping forces that are experienced as theshaft is moved and the induced voltages are allowed to produce currentin the windings which resists the shaft motion (i.e. electric braking).

Like all electric machines, embodiments described herein can produceforce in response to shaft motion. These forces are the result ofcurrents induced into the windings when the shaft is moved. The conceptof a change in magnetic fields through a winding (i.e. shaft motionthrough windings) has long been exploited as a means to transformenergy; it is used in motors as a form of braking when windings areshorted together to allow the induced currents to circulate through thewindings which in turn produces forces that oppose the motion thatcaused the change in fields.

Embodiments described herein also exhibit this property; if windingleads are shorted together (for example by connecting them both toground through an h-bridge driver), then motion of the shaft will causethe magnetic fields within the coils to change, which in turn generatesa voltage within the windings, and in turn results in current flowingthrough the winding; the force due to said current can be convenientlycalculated by multiplying said current by the winding response functionfor the winding containing the current. Additionally, the voltage (andin turn, the current) generated in a winding by shaft motion can becalculated by multiplying said shaft motion by the winding generationfunction for said winding. It follows that the force generated by awinding due to shaft motion in an embodiment, is found by multiplyingthe shaft motion by the winding generation function (which yields acurrent) and further multiplying the result (the current) by the windingresponse function. The force generated by an embodiment can be found bysumming all such winding forces due to shaft motion.

In at least most electric machines, the forces generated due to motionare not smooth; that is to say that the forces generated are notconstant with respect to position. The force ripple associated with thisresponse is commonly termed “cogging,” although this phenomenon is notthe only factor causes electric machines to “cog” or exhibit forceripple.

A similar function to the winding response function is the ‘phaseresponse function.’ The phase response function is the sum of thewinding response functions of the windings within a phase. All windingswithin a phase have similar winding response functions, in that they arephase shifted by an integer multiple of one-half the shaft period (i.e.by n*π where n is an integer). As previously discussed, when windingdiffer in phase position by π, the polarity of these windings isreversed; this ensures that their winding response functions sum and donot cancel. When functioning correctly, a phase response function isequal to the product of the number of windings in a phase multiplied byany winding response function of a winding included in that phase. Thephase response function for a phase is therefore also sinusoid-like.Examples of phase response functions appear in FIGS. 5A though 5D.

Because all the windings within a winding pack are constructed such thatthey generate a similar magnetic field in response to current, all phaseresponse functions in an embodiment have the same shape; becausewindings are spaced evenly along the central axis, their responsefunctions are translated relative to one another. It is convenient todefine a ‘characteristic response function’ (shown in the block diagramof FIG. 8) which crosses the origin and initially increases (like thesin-function), and which has the same shape of the phase responsefunctions. The phase A response function (shown in FIG. 5A) of theembodiment shown in FIGS. 2A and 2B is then said to be thecharacteristic response function for said embodiment; in this case, thephase response function for any other phase can be obtained byphase-shifting the characteristic response function by an amount equalto the phase-shift between that phase and phase A. For example, thephase response function for phase C (shown as 1C in FIGS. 2A and 2B) isequal to the characteristic response function (or the phase A responsefunction FIG. 5A), shifted by 3π/2.

Commutation Method

One commutation method combines the following configurations: thesinusoid-like shaft field function; the relationship between windingperiod, shaft period, and the number of phases; and the previouslydiscussed organization of windings into phases.

$\begin{matrix}{{\sum\limits_{n = 0}^{N - 1}{\sin^{2}\left( {\theta - \frac{n*\pi}{N}} \right)}} = \frac{N}{2}} & {{Equation}\mspace{14mu} 1}\end{matrix}$

The above is an equation representing the trigonometric relationshipexpressing the sum of N squared sinusoidal samples that are evenlydistributed across a half-period as the constant value N/2, regardlessof the phase angle used as argument.

$\begin{matrix}{T_{shaft} = {\frac{2N_{phases}}{N_{phases} - 1}T_{winding}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$

The above ratio is an equation representing a relationship between theshaft period (T_(shaft)), the number of phases (N_(phases)), and thewinding period (T_(winding)); when a motor using these geometries iscombined with a shaft having any sinusoid-like field function, a phasecommutation function can be obtained to produce a linear force response.

$\begin{matrix}{{\Delta\varphi}_{winding} = {{\frac{T_{winding}}{T_{shaft}}2\pi} = {\pi - \frac{\pi}{N_{phases}}}}} & {{Equation}\mspace{14mu} 3}\end{matrix}$

Above is an equation, in radians, for the phase-shift between any twoadjacent phases. By multiplying both sides by N_(phases), it is clearthat the total phase-shift covered by all phases in a motor constructedaccording to this ratio is an integer multiple of pi radians. This evendistribution of phases is useful in exploiting the trigonometricrelationship of Equation 1 in order to achieve linear force outputwithout relying on a force-sensor for feedback control (i.e., in orderto achieve linear force output with open-loop force control).

It is convenient to recall that magnetic fields, and magnetic forceinteractions, are subject to the superposition principal; in otherterms, the resulting force generated between a shaft and a winding packis the vector sum of the forces generated by every individual winding.

Commutating an embodiment requires a function which dictates howwindings should be selectively powered given a shaft position and givena commanded output; such functions are referred to as a phasecommutation functions. Phase commutation functions appear in blockdiagram FIG. 8. These functions are plotted as current per commandedforce output versus shaft position. When these functions are passed aphase position as argument and multiplied by the commanded force, acurrent for the associated phase is obtained; the force generated insaid phase due to said current can be calculated by multiplying saidcurrent by the result of the associated phase response function, usingthe same given phase position as an argument.

Phase commutation functions are assumed to be sinusoid-like, and similarto their associated phase response functions in terms of zero-crossings(and thus peak magnitude locations); this assumption ensures that zerocurrent is passed through the phase when the force constant is zero, andthat maximum current is passed through the phase when the force constantis maximum. A result of this assumption is that the phase commutationfunction is subject to the same convenience of phase response functions:they can be represented by a single characteristic function that isphase-shifted to obtain specific phase response functions. Thecharacteristic commutation function appears in block diagram FIG. 8.

The output of the embodiment can be determined by adding the forcegenerated by each phase.

$\begin{matrix}{{Force} = {\sum\limits_{n = 0}^{N - 1}{{{crf}\left( {\theta + {n*{\Delta\varphi}_{winding}}} \right)}*{{ccf}\left( {\theta + {n*{\Delta\varphi}_{winding}}} \right)}*C}}} & {{Equation}\mspace{14mu} 4}\end{matrix}$

The above equation illustrates this concept; elements of the summationrepresent all the phases in the motor having been commutated by applyingan appropriate phase shift of the characteristic response function (crf)and characteristic commutation function (ccf); commanded force (C) ismultiplied by every phase commutation function at the phase position (θ)which returns a current, which is multiplied by the phase responsefunction at the phase position (θ), which returns the force generated bythat phase; these forces are summed and the output of the embodiment ispredicted.

Equation 4 can be rewritten as follows:

$\begin{matrix}{{Force} = {\sum\limits_{n = 0}^{N - 1}{{{crf}\left( {\theta + {\pi n} - \frac{\pi n}{N_{phases}}} \right)}*{{ccf}\left( {\theta + {\pi n} - \frac{\pi n}{N_{phases}}} \right)}*C}}} & {{Equation}\mspace{14mu} 5}\end{matrix}$

By noting that the product of two sinusoid-like functions sharingzero-crossings and peak polarities is again periodic, and the product'speriod is half that of either sinusoid's original period, Equation 5 canbe further simplified to:

$\begin{matrix}{{C*{\sum\limits_{n = 0}^{N_{ph{ases}} - 1}{{{crf}\left( {\theta - \frac{\pi n}{N_{phases}}} \right)}*{{ccf}\left( {\theta - \frac{\pi n}{N_{phases}}} \right)}}}} = {Force}} & {{Equation}\mspace{14mu} 6}\end{matrix}$

The examples of FIGS. 7A and 7B can be used to see that if thesefunctions are multiplied together, the resulting function's period willbe 7C instead of a as the product of two negative numbers is positive.This mathematical concept follows from the fact that magneticinteraction is equal in magnitude and opposite in polarity when one ofthe sources has its polarity flipped. For example, the phase responsefunction of FIG. 5A between zero to 7C is similar to the response from πto 2π, except that the polarity is flipped. This reversing of polarityis owing to the fact that from π to 2π, a shaft field function is equalin magnitude and opposite in polarity when compared to that shaft'sfield function from zero to 2π; FIG. 3B can be used to see thisgraphically. Thus, the term of nit in Equation 5 is redundant as itrepresents an integer multiple of a period and is removed from thefunction argument.

To achieve a linear force output, regardless of the shaft position, thesimilarity between Equation 1 and Equation 6 is exploited. If thecommanded output (‘C’) is taken to be constant, as is the case when aconstant output is desired, it can be taken outside from the summation.If for every phase position, the product of the characteristic responsefunction and the characteristic commutation function is equal to theproduct of 2 multiplied by the sin²-function, then divided by N, thenEquation 6 simplifies to:

$\begin{matrix}{{{C = {Force}};}{{{assuming}:{{{crf}(\theta)}*{{ccf}(\theta)}}} = {\frac{2}{N}{\sin^{2}(\theta)}}}} & {{Equation}\mspace{14mu} 7}\end{matrix}$

It can be seen in Equation 7 above that given a constant commandedforce, regardless of the shaft position, the response of the embodimentis constant. Furthermore, the force output is linearly proportional tothe commanded force, regardless of the phase position. The properties ofsaid sin²-function are discussed in more detail below.

If follows that in order to achieve a linear force response that willnot ripple with position, a characteristic commutation function can begenerated according to the following relationship:

$\begin{matrix}{{{ccf}(\theta)} = {\frac{2}{N}*\frac{\sin^{2}(\theta)}{{crf}(\theta)}}} & {{Equation}\mspace{14mu} 8}\end{matrix}$

Corresponding phase commutation functions can be obtained via thefollowing:

pcf _(n)(θ)=ccf(θ+nΔφ _(winding))   Equation 9

FIG. 7A illustrates an embodiment's characteristic response function(thick curve 31—left y-axis) overlaid with the characteristiccommutation function (thin curve 30, right y-axis) that was found usingthis method; in this case, the embodiment's shaft did not spacepermanent magnets apart, similar to an embodiment represented by FIG.4A. FIG. 7B illustrates another embodiment's characteristic responsefunction (thick curve 33, left y-axis) overlaid with the characteristiccommutation function (thin curve 32, right y-axis) that was found usingthis method; this embodiment makes use of iron spacers and has similardimensions to an embodiment represented by FIG. 3A.

The sin²-function used in generating a commutation function from aresponse function is horizontally scaled and translated such that it haszeros which are spatially aligned with the zeros of the characteristicresponse function; in other words, the sin²-function is horizontallyscaled such that the period of the sin²function becomes one-half theperiod of the characteristic response function; for example, if such asin²-function was to be used with a characteristic response functionsimilar to FIG. 5A, the period of the sin²-function would be π. Inembodiments where the characteristic response function does not crossthe origin, the sin²-function is also translated such that the zeros ofthe two functions align. It is also assumed said sin²-function has amagnitude of 1 and it unitless. It is useful to note that thesin²-function is always positive, thus a function derived by dividingthe sin²-function by a phase response function will have values with thesame sign (i.e. positive or negative) of the phase response function atall points. It is also useful to note that a function derived bydividing such a sin²-function by a phase response function will haveunits of current per force.

When command of specific forces is used in an embodiment, one shouldkeep track of the units used when deriving the commutation functions.When a commutation function is formed from a response function, theresponse function's values (force-constants) are in unit of force perunit of current, and these same units will be used in the commutationprocess: said unit of force is used to scale the commutation function toderive said unit of current, which is passed through the correspondingwinding. In this way, specific commanded forces are realized at anyarbitrary shaft position—for example, in units of newtons.

When command of specific forces is not required (i.e. when inputs relateonly to maximum output of the device, and not a specified force), onlythe shape of the functions needs to be maintained; in other words,response functions can be normalized according to their peak magnitude,and the 2/N term in Equation 8 is removed; the result of suchnormalization will result in a commutation function having a peak valueof one. The maximum allowable input to be used (i.e. scaled bycommutation functions) is then typically defined to equal the maximumcurrent allowable in a phase. It is important to note that whenfunctions are normalized as such, only a linear output is achieved:specific knowledge of the output is not possible without furtherprofiling techniques.

Force Sensing

When an embodiment has access to the response functions that relate toits phases (e.g. if said response functions are saved to the permanentmemory of a microcomputer included in the embodiment), and further hasaccess to the amount and direction of current in its phases (e.g. theembodiment has drivers having current sensors), then the embodiment candetermine the amount of force generated. This force generation could bedue to power being selectively applied to the phases to achievecommutation, or could be due to currents induce in the phases due toshaft motion; in any case, the force generated is equal to the sum ofthe current in all phases multiplied by the corresponding phase responsefunctions.

Applicability to Rotary Motors

It is recognized also that the characterization methods disclosed hereinapply equally to permanent-magnet brushless rotary motors in producing alinear torque output. That is to say that a characteristic responsefunction can be measured or predicted, and a characteristic commutationfunction that would produce linear torque output can be easilydetermined using the discussed method, due to rotary motors' periodicnature and even distribution of phases.

Typical Implementations of Method

Post-Profiling: Text Fixture

One method for constructing an embodiment involves building or obtaininga machine with an appropriate shaft response function and relationshipsthat satisfy Equation 2 without concern for the shape of thecharacteristic response function. The characteristic response functioncan then be measured in at least two ways.

First, some known quantity of current can be applied though a phase ofan embodiment. The force output can be measured over a range of valuesat least equal to one shaft period; a single period of this response(divided by the known quantity of current applied to said phase) is thecharacteristic response function. A suitable method of accomplishingthis is by attaching a load cell between the shaft of an embodiment anda moving stage; the stage is moved while powering a phase in theembodiment and the forces are measured at a number of shaft positions.To get the corresponding characteristic commutation function, Equation 8is employed.

Post-Profiling: Current-Sense

Alternatively, owing to the previously discussed relationship betweenthe response function and the generation function, the shafts of anembodiment can be moved while the shaft speed, shaft position, andcurrent through a phase is measured. If each measured current sample isnormalized according to the speed of the shaft when it was taken, thenthe resulting normalized periodic function will have the same shape asthe characteristic response function. This method does not produce acharacteristic response function with units of force-per-current, as thegeneration function was normalized thus producing a unit-less function,but the shape can still be combined with a sin²-function according toEquation 8—the result of which can be scaled by an appropriate value toyield a characteristic commutation function that will produce a linearforce output. While this method fails to enable a known force outputfrom a commanded force without additional profiling, it enableslinearization of an embodiment with very little electronics and noprofiling fixture; this method is typically achievable using only acurrent-sense-enabled driver typically included in an embodiment.

Pre-Profiling

If an accurate model of the response function can be obtained using asoftware modeling program, this response function can be used to obtaincommutation functions via Equation 8.

Pre-Designing

Another method of constructing an embodiment involves building thedevice such that it naturally has a sine-shaped characteristic responsefunction. This can be accomplished using electromagnetic simulationintegrated into 3-D computer-aided drawing software and typically ironspacers are used when designing sinusoidal characteristic responsefunctions; typically, the ratio of iron to magnet used in a shaft ismanipulated to change and shape the characteristic response functionduring design, although other methods (e.g. using non-magnetic spacers)are also sometimes used. When an embodiment having sinusoidal responsefunctions is driven using commutation functions having the shape of asine function, its force output will be linear and will have no ripple.Two advantages of this method are that no force-measuringcharacterization process is required, and the damping response (i.e.electric braking) will be linear (discussed later).

FIGS. 1, 2A, and 2B are examples of an embodiment with nearly sinusoidalphase response functions (said phase response functions can be seen inFIGS. 5A, through 5D) that can be driven with sinusoidal commutationfunctions to achieve a good linear output; when such an embodiment isused in a feedback controller—for example, a helicopter simulatorcontroller—the linearity of the output is such that the human operatingthe simulator cannot discern the output ripple. Furthermore, when drivenby phase commutation functions shown in FIGS. 6A through 6D, which canbe obtained using the method described herein, the output can belinearized to, for example, within approximately one-percent at anyposition.

Magnet Spacing

Magnetic materials like iron can be used in between permanent magnets ina shaft to gain a number of advantages and change the way an embodimentperforms. Magnetic materials such as iron are said to have magneticdomains which can be described as regions within the material that aremagnetized in a uniform direction; this means that the individualmagnetic moments of the atoms are aligned with one another and theypoint in the same direction. The direction in which domains align dependentirely on the vector sum of fields within the domain; the vector sumof fields is due in part to neighboring domains, induced fields due tocurrent passing through nearby windings, permanent magnets in theproximity of said domains, and other magnetic phenomena. The fields ofthe iron or other magnetic material will constructively interfere withthe shaft's field, resulting in a greater field magnitude within andimmediately surrounding the material. Magnetic material used as spacersthus results in a shaft field function curve with a different shape thanif non-magnetic spacers, or no spacers (back-to-back adjacent magnets asin FIG. 4A) were used in an embodiment. In general, the addition of ironspacers between permanent magnets results in a shaft field function thatis smoother—or at all positions is changing values more gradually—thanif a shaft is constructed purely of alternating permanent magnets. FIGS.3 and 4 illustrate two shafts of different construction and theresulting shaft field function.

Iron Advantage: Easier Manufacture

During assembly, magnetic spacing materials can cause thealternating-polarity permanent magnets to stay in place, stuck to thespacer, instead of repelling each other and requiring significantassembly forces. FIGS. 9A through 9D are used to illustrate thisconcept; in these sectional views, hashmarks are used to indicate bothmaterial type (hash spacing) and magnetic domain alignment (hash angle);for example, materials with parallel hashmarks share domain alignment.Force vectors beside 52 are included to depict the (axial) forceexperienced during assembly; forces experienced by magnet 50 and iron 51are not shown. When a magnet (50) is placed near an iron spacer (51) asin FIG. 9A, the iron spacer aligns its domains with the magnet and aforce is generated attracting the two. As a second magnet (52) orientedopposite the first approaches the iron spacer (51), initially arepulsive force is experienced (also shown in Equation 8). As magnet 52is forced toward spacer 51, magnetic fields from 52 will overpower themagnetic fields from magnet 50 in a portion of iron 51 as shown in FIG.9B. More domains in iron 51 change polarity as the distance between iron51 and magnet 52 decrease; the force experienced by magnet 52 in turndecreases and eventually also changes polarity as shown by FIG. 9C. Whenthe three components are connected as in FIG. 9D, they all experienceattraction. Iron spacer 51 experiences a repulsion force between itsdomains of opposite polarity, but atomic forces are significantly higherthan the magnetic-repulsion force and, therefore, hold the materialtogether.

Embodiments having magnets spaced adjacent with no spacers, as in FIG.4A, will exhibit extremely high assembly forces which may necessitatespecial machines for production. In the event a shaft is damaged ordisassembled, the contents of said shaft may come apart, acceleratingrapidly, and pose a danger to machines and people nearby. Embodimentshaving sufficient magnetic material in between permanent magnets suchthat the local domain forces overpower the repulsive forces of thenearby magnets (as in FIG. 9D) can be assembled by hand, and would notfly apart if the shaft were damaged or disassembled.

Iron Advantage: Improved Effectiveness

When designed correctly, iron spacers between permanent magnets can be acost-effective method of increasing the efficiency at which current isconverted to force, and shaft motion is converted to current. Becauseiron is a significantly less expensive commodity than the neodymiumtypically used in electric machines, a shaft's field function can beincreased in magnitude without significantly increasing the cost ofmaterials for an embodiment, for a given volume of neodymium used. Sincethe effectiveness of transforming current into forces or shaft motioninto current in a LEM is dependent (among other things) on the fluxwithin the shaft, increasing the magnitude of the shaft field functionby using iron spacers increases the effectiveness of a machine's energyconversion, without incurring the costs of higher neodymium volumes. Anexample of improving an embodiment's performance-to-cost ratio by usingiron spacers can be described using FIGS. 3A and 4A. The material costsof the shaft depicted in FIG. 3A will be typically between half totwo-thirds the costs of the shaft detailed in FIG. 4A. However, theperformance of the two devices is more similar in nature—typicallyhaving the embodiment of FIG. 3A generating between seventy percent toninety percent of the forces of the embodiment of FIG. 4A, for a givenamount of electrical power. In other words, adding iron or othermagnetic materials into a given shaft dimension allows design ofembodiments to achieve different cost-to-performance ratios. It isrecognized that this advantage also applies equally to embodiments usedfor motion-to-electrical energy generation, and equally to embodimentsused for both directions of energy conversion.

Iron Advantage: Faster Linear Force Output

As previously discussed, magnetic spacer materials tend to create asmoother shaft field function. Smoother shaft field functions result ina more gradually changing imbalance of magnetic fields on either side ofa winding as the shaft position is changed; this in turn yields a phaseresponse function that changes more gradually, which typically yields acommutation function that changes more gradually. The inductive natureof windings places a limit on the rate at which current can change withthe windings—the maximum rate typically being a function of the voltageavailable to the drivers. Thus, an embodiment having a smoothercharacteristic commutation function (and all other things being equal),will be enabled to move at a higher rate while sustaining linear outputthan another embodiment having “sharper” characteristic commutationfunctions.

Iron Advantage: Linear Damping Response

As discussed, LEMs and all electric machines can produce force whensubject to motion. As an embodiment's shaft moves relative to a windingpack, a change of flux through the windings results in a voltage, andconsequently a current to flow through the windings. When constantmotion is applied to the shaft (or winding pack), the voltages inducedinto the phase windings form a function versus the shaft position; thesefunctions were previously described as phase generation functions, andit was discussed that these functions are identical in shape (oncenormalized) to the phase response functions. As discussed, the resultingforce due to shaft motion can be obtained by summing the forcesgenerated in each winding, or specifically, by multiplying the shaftmotion, the winding (or phase) response function, and the winding (orphase) generation function for every winding (or phase), and summing theresults.

An advantage of constructing a shaft that produces a characteristicresponse function shape that is very close to the sine-function, is thatmotion of the shaft will then generate sinusoidal current waveforms inthe phases (because a sinusoidal response function implies a sinusoidalgeneration function), which when multiplied by the sinusoidal responsewaveforms will take the shape of a sin²-function; in other words, asinusoidal characteristic response function means that constant shaftmotion will cause the phases to generate sin²-shaped forces (opposingsaid motion). Due to the previously discussed result of the sum ofevenly-distributed sin²-functions (shown mathematically by the equationof Equation 1), the force produced as a result of shaft motion is linear(i.e. has relatively low ripple). A term for this result is ‘lineardamping response’ and can be considered as ‘linearized cogging.’

While an embodiment with very “sharp” response functions can obtainlinearized output using the commutation methods provided herein, saidembodiment's damping response is a function of its construction alone.

Iron Advantage: Reduced Sensor Resolution Requirements

The phase-position resolution requirements of an embodiment depend onthe geometry and required performance of an embodiment; in general,commutation functions that deviate greatly from a sine-function (i.e.“sharper” functions) may require higher-resolution position sensing thanthose commutation functions that are smoother. Thus, another advantageof including iron in an embodiment and achieving characteristiccommutation functions close in shape to a sine wave, is thatshaft-position-information resolution can be lower. Exact resolutionrequirements depend on implementation, but in general, if anembodiment's characteristic commutation can be made smoother thananother embodiment's, the position resolution required to describe thesame delta of commutation function (i.e. delta of phase current) betweenany two adjacent measurable shaft positions, will be decreased(improved) by the former embodiment. As the delta of phase currentsbetween any two measurable shaft positions increases, commutationbetween those positions will result in an increasing delta in forceoutput: a “tick” or a step function in force output; determining theminimum sensor resolution requirements involves defining the minimum“tick” or step that's acceptable. Given the characteristic commutationfunction, the minimum step in commutation function will define theminimum sensor resolution.

Iron Drawback: Saturation Requirements

When ferrous spacing material is used, the shaft is configured to ensurethat the field strength within the spacer material from the adjacentmagnets is greater than the maximum field strength generated by thewindings during operation; if this is not observed, a significantpercentage of the domains of the spacer material can change duringoperation of an embodiment. If the domains of the magnetic spacermaterial change due to the commutated winding fields, the shaft fieldfunction changes with them. In at least one advantageous method ofcommutation discussed herein, linear output of an embodiment isdependent on knowledge of the relationships between the shaft andwinding pack; if these relationships are altered—as changing the shaftfield function would achieve—the linearity of the output could becompromised

Non-Magnetic Magnet Spacers

Magnets also can be separated from one another using non-magneticmaterials like plastic. Like the method of adding magnetic spacermaterial, using non-magnetic spacers can result in a commutationfunction that is closer to a pure sine wave (i.e. smoother), as theshaft field function will change directions over a wider range ofvalues. Configuring magnets to be spaced apart by non-magnetic materialscan achieve a linear damping response as described above and can improvethe maximum speed at which a linear response is possible as describedabove. Configuring magnets to be spaced apart by non-magnetic materialswill decrease the assembly forces required to construct the shaft, whencompared to when magnets are configured to be touching each other.

Typical Operation

Typically, the commanded output and phase position are received by amicrocontroller (represented in FIG. 8 by box 24); the microcontroller,having the phase commutation functions saved to its permanent memory, orhaving the characteristic commutation function and the number of phasessaved to permanent memory such that it can derive the phase commutationfunctions via Equation 9, combines the phase commutation function withthe phase position and multiplies the results by the commanded output toreceive the respective phase currents; the microcontroller thenmanipulates the drivers to realize these currents within the windings,thus producing the commanded output.

Often, functions like the characteristic commutation function are savedto the memory of a microcomputer in the form of a lookup table that isindexed by the phase position, or by the result of some math involvingthe phase position.

It is recognized that a microcontroller, or other entity capable ofperforming the commutation method, may control current to the windingsin a number of ways, including by using applied voltages (or h-bridgeduty cycles), or alternatively by using feedback from current sensorsand some closed-loop control method—typically aproportional-plus-integral control method.

Because the forces generated are entirely dependent on the currentwithin the windings, and because current does not change instantly whensubject to an applied voltage, using voltage control alone to realizethe commutation currents (e.g. by multiplying the required current bythe phase resistance and applying the result in the form of a voltage),can degrade the linearity of the response, as the resulting currents arenot realized instantly. Often this problem is improved upon by usingcurrent sensors that generate a signal that is used in closed-loopcurrent control to reduce the realization time of the commutationcalculations. Current control also ensures that as windings heat up(possibly unevenly), the current delivered them to does not decrease.The embodiments described herein are typically driven and controlledusing these and other methods common to the driving and controlling ofother electric machines.

Due to the simplicity of materials and the relative ease of constructionwhen compared to existing technologies, as well as the performanceresulting from the speed and lack of friction characteristic of magneticfields, embodiments described herein are well suited to force-feedbackdevices such as simulation controls, remote controls, and gamingcontrols. FIGS. 11A through 11D and FIGS. 12A and 12B can be used toillustrate typical operation of a two-phase embodiment being attached toa linear throttle control. The throttle is intended to exhibit dampingthroughout its travel, to exhibit stiction when the speed is zero, toexhibit a “detent” centered at seventy-five percent of its maximumtravel, and to exhibit a spring force beyond eighty-percent of itsmaximum travel. FIGS. 11A through 11D are aligned according to the timeaxis and represent a user pushing the throttle from zero-percent toone-hundred-percent in one smooth motion over three seconds. FIG. 11Arepresents the shaft's position at any point in time, and its straightline indicates a constant movement, despite the forces being output fromthe embodiment (this unrealistic scenario is used to simplifyillustration). FIG. 11B represents the force commanded of theembodiment—typically from a system connected to both the embodiment andwhatever the throttle is controlling. A stiction force can be seenresisting the motion at time zero. While the shaft is in constant motionfrom time zero to three-seconds, there is constant negative dampingforce. After 2 seconds, there is a “detent” force, after which a springforce increases as the shaft position moves away from eighty-percent.FIG. 11C represents the current provided to one of two phases in theembodiment (solid line), while the phase response function for thatphase is overlaid for illustration (dashed line), and FIG. 11Drepresents the current provided to the other phase (solid line) and thephase response function of that phase (dashed line); these currents aretypically controlled via h-bridges by using current sensors andclosed-loop control; often it is advantageous to include amicrocontroller within the embodiment that can be configured to receivecurrent sense information and perform close-loop current control via theh-bridges to realize the phase currents that are calculated via thecommutation functions. It is understood that the solid lines in FIGS.11C and 11D are the result of real-time control of the current withinthe phases, while the dashed lines are the result of the staticrelationships between the shaft and phases in this particularembodiment; it is further understood that the dashed lines weredetermined during design or during a profiling step and were then usedto determine the commutation functions that, when combined with thecommanded force and the phase position, were used to determine therequired (and thus realized) currents in the phases (solid lines). Thecurrent applied to the two phases results in a force generated betweenthe shaft and the phases; FIG. 12A illustrates the force output of phaseA (curve 41) and the force output of phase B (curve 40); FIG. 12Bfurther illustrates these functions of force output versus time, whilealso including their sum: the force output of the embodiment (curve 42).The force curves, 41 and 40, follow from the current curves (solidlines) of FIGS. 11C and 11D respectively, multiplied by the respectivephase response curves (dashed lines).

From the foregoing it will be appreciated that, although specificembodiments have been described herein for purposes of illustration,various modifications may be made without deviating from the spirit andscope of the disclosure. Furthermore, where an alternative is disclosedfor a particular embodiment, this alternative may also apply to otherembodiments even if not specifically stated. Furthermore, one or morecomponents of a described apparatus or system may have been omitted fromthe description for clarity or another reason. Moreover, one or morecomponents of a described apparatus or system that have been included inthe description may be omitted from the apparatus or system.

1-24. (canceled)
 25. A linear electric machine, comprising: one or more shafts including one or more magnets configured to produce a sinusoid-like magnetic-field pattern; one or more winding packs, each defining a central bore and a central axis; each winding pack being configured to receive one of the one or more shafts along a central axis and within the defined central bore, each winding pack including two or more phases of windings being distributed along the shared central axis, each phase spaced apart from an adjacent phase by an approximately same interval, the product of said interval multiplied by the number of said phases equal to an integer multiple of one-half the period of the sinusoid-like magnetic-field pattern, the number of phases being more than three; and a phase-position sensor configured to detect the alignment of the shaft magnetic field with respect to at least one phase.
 26. The linear electric machine of claim 25, wherein the shaft contains a magnetic material in between the magnets.
 27. The linear electric machine of claim 25, wherein the shaft magnets are spaced by non-magnetic material.
 28. The linear electric machine of claim 25, further comprising spacers, each disposed between a respective pair of windings.
 29. The linear electric machine of claim 28, wherein the spacers contain features for routing the leads from the windings.
 30. The linear electric machine of claim 28, wherein the spacers contain features for mounting electronic components.
 31. The linear electric machine of claim 28, wherein the spacers each include a respective printed-circuit assembly.
 32. The linear electric machine of claim 25, further comprising h-bridge circuits each configured to power a respective one of the phases selectively.
 33. The linear electric machine of claim 25, further comprising a microcomputer coupled to the phase-position sensor, including a memory storing at least one communication function, and configured to realize a force command by selectively powering the phases in response to at least one of the at least one communication function and an output signal of the phase-position sensor.
 34. The linear electric machine of claim 33, wherein the microcomputer memory also stores one or more response functions, and where the microcomputer is configured to translate measured currents within the phases into corresponding forces, and to determine the amount of force generated between the at least one shaft and the at least one winding pack.
 35. The linear electric machine of claim 25, wherein the magnetic fields of the shaft are such that the characteristic response function of the LEM, when normalized according to a maximum magnitude thereof, and when scaled and translated along an x-axis such that it shares zero-crossings and polarity with a sin-function, falls within ten-percent of the sin-function's value for all phase angles.
 36. A method for profiling a linear electric machine including a shaft and winding phases, wherein: a normalized function of force-per-current constants versus position for one or more phase is obtained, comprising the steps of: changing a respective position of the shaft relative to the winding phases; measuring resulting phase currents and a relative speed between the winding phase and the shaft for respective positions of the shaft relative to the winding phases; and determining the normalized function of force-per-current constants at said respective positions by dividing said phase currents by said speeds.
 37. The method of claim 36, wherein a test-fixture empirically measures force-per-current constants of winding phases at respective positions of the shaft relative to the phases.
 38. The method of claim 36, wherein a computing circuit executing magnetic interaction modelling algorithms analytically predicts force-per-current constants of winding phases at respective positions of the shaft relative to the phases.
 39. The method of claim 36, whereby the LEM is enabled to exert predictable forces via commutation functions which are derived from the functions of force-per-current versus relative position of the shaft with respect to the phases.
 40. A system comprising: one or more components; and a linear electric machine, including: one or more shafts each including one or more magnets configured to produce a sinusoid-like magnetic-field pattern, at least one of the one or more shafts coupled to at least one of the one or more components; one or more winding packs, each defining a central bore and a central axis; each winding pack being configured to receive one of the one or more shafts along a central axis and within the defined central bore; each winding pack including two or more phases of windings being distributed along the shared central axis, each phase spaced apart from an adjacent phase by an approximately same interval, the product of said interval multiplied by the number of said phases equal to an integer multiple of one-half the period of the sinusoid-like magnetic-field pattern, the number of phases being more than three; and a phase-position sensor configured to detect the alignment of the shaft magnetic field with respect to at least one phase.
 41. The system of claim 40 wherein at least one of the one or more components is a mechanical ground.
 42. The system of claim 40 wherein at least one of the one or more components is a manipulator configured to be manipulated by a human.
 43. The system of claim 40 wherein: at least one of the one or more components is a remotely-located machine; and the position of at least one of the one or more shafts included in the LEM is used to manipulate the state of the remotely-located machine.
 44. The system of claim 40 wherein: at least one of the one or more components is a remotely-located machine; and the force output of the LEM is used to impart information relating to the state of the remotely-located machine.
 45. The system of claim 40 wherein: at least one of the one or more components is an electronic circuit configured to simulate a connection to a machine; and the force output of the LEM is used to impart information relating to the state of the simulated connection. 